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Diversity-aware clustering: Computational Complexity and Approximation Algorithms (2401.05502v3)
Published 10 Jan 2024 in cs.DS, cs.AI, cs.CC, and cs.LG
Abstract: In this work, we study diversity-aware clustering problems where the data points are associated with multiple attributes resulting in intersecting groups. A clustering solution needs to ensure that the number of chosen cluster centers from each group should be within the range defined by a lower and upper bound threshold for each group, while simultaneously minimizing the clustering objective, which can be either $k$-median, $k$-means or $k$-supplier. We study the computational complexity of the proposed problems, offering insights into their NP-hardness, polynomial-time inapproximability, and fixed-parameter intractability. We present parameterized approximation algorithms with approximation ratios $1+ \frac{2}{e} + \epsilon \approx 1.736$, $1+\frac{8}{e} + \epsilon \approx 3.943$, and $5$ for diversity-aware $k$-median, diversity-aware $k$-means and diversity-aware $k$-supplier, respectively. Assuming Gap-ETH, the approximation ratios are tight for the diversity-aware $k$-median and diversity-aware $k$-means problems. Our results imply the same approximation factors for their respective fair variants with disjoint groups -- fair $k$-median, fair $k$-means, and fair $k$-supplier -- with lower bound requirements.
- Healey, J.F., Stepnick, A.: Diversity and Society: Race, Ethnicity, and Gender. Sage Publications, London, UK (2019) Shore et al. [2018] Shore, L.M., Cleveland, J.N., Sanchez, D.: Inclusive workplaces: A review and model. Human Resource Management Review 28(2), 176–189 (2018) Zinn and Dill [1996] Zinn, M.B., Dill, B.T.: Theorizing difference from multiracial feminism. Feminist Studies 22(2), 321–331 (1996) Crenshaw [2013] Crenshaw, K.: Demarginalizing the intersection of race and sex: A black feminist critique of antidiscrimination doctrine, feminist theory and antiracist politics, 23–51 (2013) Runyan [2018] Runyan, A.S.: What is intersectionality and why is it important? Academe 104(6), 10–14 (2018) Fish [1993] Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Shore, L.M., Cleveland, J.N., Sanchez, D.: Inclusive workplaces: A review and model. Human Resource Management Review 28(2), 176–189 (2018) Zinn and Dill [1996] Zinn, M.B., Dill, B.T.: Theorizing difference from multiracial feminism. Feminist Studies 22(2), 321–331 (1996) Crenshaw [2013] Crenshaw, K.: Demarginalizing the intersection of race and sex: A black feminist critique of antidiscrimination doctrine, feminist theory and antiracist politics, 23–51 (2013) Runyan [2018] Runyan, A.S.: What is intersectionality and why is it important? Academe 104(6), 10–14 (2018) Fish [1993] Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zinn, M.B., Dill, B.T.: Theorizing difference from multiracial feminism. Feminist Studies 22(2), 321–331 (1996) Crenshaw [2013] Crenshaw, K.: Demarginalizing the intersection of race and sex: A black feminist critique of antidiscrimination doctrine, feminist theory and antiracist politics, 23–51 (2013) Runyan [2018] Runyan, A.S.: What is intersectionality and why is it important? Academe 104(6), 10–14 (2018) Fish [1993] Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Crenshaw, K.: Demarginalizing the intersection of race and sex: A black feminist critique of antidiscrimination doctrine, feminist theory and antiracist politics, 23–51 (2013) Runyan [2018] Runyan, A.S.: What is intersectionality and why is it important? Academe 104(6), 10–14 (2018) Fish [1993] Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Runyan, A.S.: What is intersectionality and why is it important? Academe 104(6), 10–14 (2018) Fish [1993] Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). 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SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zinn, M.B., Dill, B.T.: Theorizing difference from multiracial feminism. Feminist Studies 22(2), 321–331 (1996) Crenshaw [2013] Crenshaw, K.: Demarginalizing the intersection of race and sex: A black feminist critique of antidiscrimination doctrine, feminist theory and antiracist politics, 23–51 (2013) Runyan [2018] Runyan, A.S.: What is intersectionality and why is it important? Academe 104(6), 10–14 (2018) Fish [1993] Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Crenshaw, K.: Demarginalizing the intersection of race and sex: A black feminist critique of antidiscrimination doctrine, feminist theory and antiracist politics, 23–51 (2013) Runyan [2018] Runyan, A.S.: What is intersectionality and why is it important? Academe 104(6), 10–14 (2018) Fish [1993] Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Runyan, A.S.: What is intersectionality and why is it important? Academe 104(6), 10–14 (2018) Fish [1993] Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. 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PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Runyan, A.S.: What is intersectionality and why is it important? Academe 104(6), 10–14 (2018) Fish [1993] Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. 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[2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). 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[2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fish, S.: Reverse racism, or how the pot got to call the kettle black. Atlantic Monthly 272(5), 128–136 (1993) Kearns et al. [2019] Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). 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ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. 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[2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kearns, M., Neel, S., Roth, A., Wu, Z.S.: An empirical study of rich subgroup fairness for machine learning. In: FAccT, pp. 100–109 (2019) Kasy and Abebe [2021] Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kasy, M., Abebe, R.: Fairness, equality, and power in algorithmic decision-making. In: FAccT, pp. 576–586 (2021) Ghosh et al. [2021] Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. 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ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. 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Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. 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[2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Ghosh, A., Genuit, L., Reagan, M.: Characterizing intersectional group fairness with worst-case comparisons. In: Artificial Intelligence Diversity, Belonging, Equity, and Inclusion, pp. 22–34 (2021). PMLR Hoffmann [2019] Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Hoffmann, A.L.: Where fairness fails: data, algorithms, and the limits of antidiscrimination discourse. Information, Communication & Society 22(7), 900–915 (2019) Kong [2022] Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kong, Y.: Are “Intersectionally Fair” AI Algorithms Really Fair to Women of Color? A Philosophical Analysis. In: FAccT, pp. 485–494 (2022). ACM Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). 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[1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM 33(3), 533–550 (1986) Guha and Khuller [1998] Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: SODA, pp. 649–657. SIAM, USA (1998) Cohen-Addad et al. [2019] Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Kumar, A., Lee, E., Li, J.: Tight fpt approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. In: ICALP (2019). Dagstuhl Fomin et al. [2022] Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. 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Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. 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In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Fomin, F.V., Golovach, P.A., Inamdar, T., Purohit, N., Saurabh, S.: Exact Exponential Algorithms for Clustering Problems. In: IPEC. LIPIcs, vol. 249, pp. 13–11314. Dagstuhl, Dagstuhl, Germany (2022) Charikar et al. [1999] Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. 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[2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. 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(2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem (extended abstract). In: STOC, pp. 1–10. ACM, New York, USA (1999) Arya et al. [2001] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k𝑘kitalic_k-median and facility location problems. In: STOC, pp. 21–29 (2001) Li [2016] Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Li, S.: Approximating capacitated k𝑘kitalic_k-median with (1+ϵ)k1italic-ϵ𝑘(1+\epsilon)k( 1 + italic_ϵ ) italic_k open facilities. In: SODA, pp. 786–796 (2016). SIAM Goyal and Jaiswal [2023] Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. 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In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Goyal, D., Jaiswal, R.: Tight fpt approximation for socially fair clustering. Information Processing Letters 182, 106383 (2023) Hajiaghayi et al. [2012] Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63, 795–814 (2012) Kleindessner et al. [2019] Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). 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[2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kleindessner, M., Awasthi, P., Morgenstern, J.: Fair k𝑘kitalic_k-center clustering for data summarization. In: ICML, pp. 3448–3457 (2019). PMLR Ghadiri et al. [2021] Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
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Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ghadiri, M., Samadi, S., Vempala, S.: Socially fair k𝑘kitalic_k-means clustering. In: FAccT, pp. 438–448 (2021) Chen et al. [2016] Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. 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In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. 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[2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. 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In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. 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In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. 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Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Chen, D.Z., Li, J., Liang, H., Wang, H.: Matroid and knapsack center problems. Algorithmica 75(1), 27–52 (2016) Krishnaswamy et al. [2011] Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. 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(2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: The matroid median problem. In: SODA (2011) Thejaswi et al. [2021] Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Thejaswi, S., Ordozgoiti, B., Gionis, A.: Diversity-aware k𝑘kitalic_k-median: clustering with fair-center representation. In: Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 765–780 (2021). Springer Thejaswi et al. [2022] Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Thejaswi, S., Gadekar, A., Ordozgoiti, B., Osadnik, M.: Clustering with fair-center representation: parameterized approximation algorithms and heuristics. In: SIG-KDD, pp. 1749–1759 (2022) Charikar et al. [2002] Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Charikar, M., Guha, S., Tardos, Shmoys, D.B.: A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. JCSS 65(1), 129–149 (2002) Arya et al. [2004] Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. 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In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k𝑘kitalic_k-median and facility location problems. SIAM Journal on computing 33(3), 544–562 (2004) Cohen-Addad et al. [2022] Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. 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(2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for k𝑘kitalic_k-median. In: SODA, pp. 1556–1612 (2022). SIAM Byrka et al. [2014] Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. 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[2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for k𝑘kitalic_k-median, and positive correlation in budgeted optimization. In: SODA, pp. 737–756 (2014). SIAM Kanungo et al. [2004] Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. 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Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k𝑘kitalic_k-means clustering. Computational Geometry 28(2-3), 89–112 (2004) Ahmadian et al. [2019] Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k𝑘kitalic_k-means and euclidean k𝑘kitalic_k-median by primal-dual algorithms. SIAM Journal on Computing 49(4), 17–97 (2019) Gonzalez [1985] Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical computer science 38, 293–306 (1985) Hsu and Nemhauser [1979] Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hsu, W.-L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979) Feldmann and Marx [2020] Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. 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In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. 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Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. 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In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. 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In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
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Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Feldmann, A.E., Marx, D.: The parameterized hardness of the k𝑘kitalic_k-center problem in transportation networks. Algorithmica 82, 1989–2005 (2020) Chierichetti et al. [2017] Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. Advances in neural information processing systems 30 (2017) Chen et al. [2019] Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Chen, X., Fain, B., Lyu, L., Munagala, K.: Proportionally fair clustering. In: ICML, pp. 1032–1041 (2019). PMLR Ahmadian et al. [2020] Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Ahmadian, S., Epasto, A., Kumar, R., Mahdian, M.: Fair correlation clustering. In: ATSTATS, pp. 4195–4205 (2020). PMLR Bandyapadhyay et al. [2019] Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
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[2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bandyapadhyay, S., Inamdar, T., Pai, S., Varadarajan, K.: A constant approximation for colorful k𝑘kitalic_k-center. In: ESA, vol. 144, pp. 12–11214. Dagstuhl, Dagstuhl, Germany (2019) Abbasi et al. [2021] Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
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[2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: FAccT, pp. 504–514 (2021) Chhabra et al. [2021] Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Chhabra, A., Masalkovaitė, K., Mohapatra, P.: An overview of fairness in clustering. IEEE Access 9, 130698–130720 (2021) Brubach et al. [2022] Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Brubach, B., Chakrabarty, D., Dickerson, J.P., Seyed Esmaeili, M.K., Knittel, M., Morgenstern, J., Samadi, S., Srinivasan, A., Tsepenekas, L.: Fairness in clustering. https://www.fairclustering.com/ (2022) Krishnaswamy et al. [2018] Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k𝑘kitalic_k-median and k𝑘kitalic_k-means with outliers via iterative rounding. In: STOC, pp. 646–659 (2018) Jones et al. [2020] Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Jones, M., Nguyen, H., Nguyen, T.: Fair k𝑘kitalic_k-centers via maximum matching. In: ICML, pp. 4940–4949 (2020). PMLR Chen et al. [2024] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science 983, 114305 (2024) Hébert-Johnson et al. [2018] Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. 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In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. 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Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Hébert-Johnson, U., Kim, M., Reingold, O., Rothblum, G.: Multicalibration: Calibration for the (computationally-identifiable) masses. In: ICML, pp. 1939–1948 (2018). PMLR Kim et al. [2018] Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Kim, M., Reingold, O., Rothblum, G.: Fairness through computationally-bounded awareness. Advances in neural information processing systems 31 (2018) Gopalan et al. [2022] Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Gopalan, P., Kim, M.P., Singhal, M.A., Zhao, S.: Low-degree multicalibration. In: Conference on Learning Theory, pp. 3193–3234 (2022). PMLR Garey and Johnson [2002] Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. 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In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. 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In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
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International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. 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In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Garey, M.R., Johnson, D.S.: Computers and Intractability vol. 29. W. H. Freeman and Co., USA (2002) Bartal [1996] Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996). IEEE Downey and Fellows [2013] Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity vol. 4. Springer, New York, USA (2013) Impagliazzo and Paturi [2001] Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. 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In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. 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(2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
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In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. 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(2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Impagliazzo, R., Paturi, R.: On the complexity of k𝑘kitalic_k-sat. JCSS 62(2), 367–375 (2001) S. et al. [2019] S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- S., K.C., Laekhanukit, B., Manurangsi, P.: On the parameterized complexity of approximating dominating set. Journal of ACM 66(5) (2019) Zhao et al. [2023] Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Zhao, A., Zhou, Y., Liu, Q.: Improved approximation algorithms for matroid and knapsack means problems. International Journal of Foundations of Computer Science, 1–21 (2023) Chen et al. [2023] Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Chen, X., Ji, S., Wu, C., Xu, Y., Yang, Y.: An approximation algorithm for diversity-aware fair k𝑘kitalic_k-supplier problem. Theoretical Computer Science, 114305 (2023) Cohen-Addad et al. [2021] Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Cohen-Addad, V., Saulpic, D., Schwiegelshohn, C.: A new coreset framework for clustering. In: Proceedings of the ACM SIGACT Symposium on Theory of Computing, pp. 169–182 (2021) Feldman and Langberg [2011] Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Feldman, D., Langberg, M.: A unified framework for approximating and clustering data. In: STOC, pp. 569–578. ACM, New York, USA (2011) Manurangsi [2020] Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Manurangsi, P.: Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In: SODA, pp. 62–81. SIAM, USA (2020) Calinescu et al. [2011] Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing 40(6), 1740–1766 (2011) Chen et al. [2022] Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In: FOCS, pp. 612–623 (2022) Hochbaum and Shmoys [1986] Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. Journal of ACM 33(3), 533–550 (1986) Cygan et al. [2015] Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015) Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
- Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, ??? (2015)
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